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Isotropy sets of periodic geodesic flows on three dimensional manifolds

  • Jakob Dittmer
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Abstract

A classical theorem of Wadsley states that if all geodesics on a Riemannian manifold M are periodic, then they have a common period, and the geodesic flow induces an S^1-action on the unit tangent bundle. Moreover, a conjecture of Berger asserts that this action is free whenever M is simply connected. In this paper, we investigate the Berger conjecture in dimension three by studying the isotropy set of the S^1-action induced by the geodesic flow. Using Morse theory for the energy functional on the free loop space of M, we establish two results in this direction. First, we show that all isotropy groups of the geodesic flow have odd order. Second, we prove that the geodesic flow either acts freely on the unit tangent bundle T1M or has at least two singular orbit types. This paper serves as preparation for the subsequent paper The Berger conjecture for three-dimensional manifolds, in which the conjecture is proved in dimension three.

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Cite

BibTeXRIS
BibTeX
@misc{dittmer2026isotropy,
  title = {{Isotropy sets of periodic geodesic flows on three dimensional manifolds}},
  author = {Jakob Dittmer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29017},
  eprint = {2610.29017},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29017},
}
RIS
TY  - UNPB
TI  - Isotropy sets of periodic geodesic flows on three dimensional manifolds
AU  - Jakob Dittmer
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29017
UR  - https://arxiv.org/abs/2610.29017
AB  - A classical theorem of Wadsley states that if all geodesics on a Riemannian manifold M are periodic, then they have a common period, and the geodesic flow induces an S^1-action on the unit tangent bundle. Moreover, a conjecture of Berger asserts that this action is free whenever M is simply connected. In this paper, we investigate the Berger conjecture in dimension three by studying the isotropy set of the S^1-action induced by the geodesic flow. Using Morse theory for the energy functional on the free loop space of M, we establish two results in this direction. First, we show that all isotropy groups of the geodesic flow have odd order. Second, we prove that the geodesic flow either acts freely on the unit tangent bundle T^1M or has at least two singular orbit types. This paper serves as preparation for the subsequent paper The Berger conjecture for three-dimensional manifolds, in which the conjecture is proved in dimension three.
ER  -